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Stream: learning: questions

Topic: Commutative rings with additional functions

Madeleine Birchfield (Jan 21 2025 at 23:59):

Let $R$ be a commutative ring with an additional function $f:R \to R$. Is the initial such commutative ring $R$ countable?

Mike Shulman (Jan 22 2025 at 00:19):

Surely the initial algebra of any finitary algebraic theory is countable (i.e., finite or countably infinite). At least in classical mathematics.

Kevin Carlson (Jan 22 2025 at 01:44):

I'd be impressed if there were a nonclassical logic in which such an initial algebra may exist without being countable, though I have very little intuition for weak constructive logics. You just apply all the operations to everything you have available and iterate that for countably many stages...do you need dependent choice or something to show this converges in countably many steps?

Mike Shulman (Jan 22 2025 at 02:45):

For a theory without axioms, definitely. But for a theory with axioms, you have to quotient the result, so it depends on what kind of [[countable]] you mean -- it'll still admit a surjection from $\mathbb{N}$, which the nLab calls "denumerably indexed", but it's not clear to me that it would admit a bijection or injection to $\mathbb{N}$.

Kevin Carlson (Jan 22 2025 at 17:55):

Ah, right, thanks.